For this, it is obviuos that for every \(0\le t \le k_i\) , \(\mathbb {Z}_{p_{i}^{\alpha _i}}^{(t)}\)  is a direct summand of \(\mathbb {Z}_{p_{i}^{\alpha _i}}^{(k_{i})}\)  and for each \(0 \le t \ne t^{\prime } \le k_i\) , we have \(\mathbb {Z}_{p_{i}^{\alpha _i}}^{(t)} \lnot \cong \mathbb {Z}_{p_{i}^{\alpha _i}}^{(t^{\prime })}\) . Now, suppose that \(C\)  is a direct summand of \(\mathbb {Z}_{p_{i}^{\alpha _i}}^{(k_{i})}\) . There exists a subgroup \(D\)  of \(\mathbb {Z}_{p_{i}^{\alpha _i}}^{(k_{i})}\)  such that \(\mathbb {Z}_{p_{i}^{\alpha _i}}^{(k_{i})} \cong C \oplus D\) . By [1]}, \(C\)  is a finitely generated abelian group. Suppose that \(C\cong \mathbb {Z}_{q_{1}^{\beta _1}}^{(l_{1})}\oplus \cdots \oplus \mathbb {Z}_{q_{s}^{\beta _s}}^{(l_{s})}\) . Since \(C\)  is a direct summand of \(\mathbb {Z}_{p_{i}^{\alpha _i}}^{(k_{i})}\)  and for every \(1\le j \le s\) , \(\mathbb {Z}_{q_{j}^{\beta _j}}^{(l_{j})}\)  is a direct summand of \(C\) , so for every \(1\le j\le s\) , \(\mathbb {Z}_{q_{j}^{\beta _j}}^{(l_{j})}\)  is a direct summand of \(\mathbb {Z}_{p_{i}^{\alpha _i}}^{(k_{i})}\) . Now, by uniqueness of decomposition of finitely generated abelian groups [1]}, for any \(j=1,\cdots ,s\) , there exists an \(i=1,\cdots ,n\)  such that \(q_j =p_i\)  and \(\beta _j =\alpha _i\) . Hence, \(C\cong \mathbb {Z}_{p_{i}^{\alpha _i}}^{(t)}\)  for some \(0\le t\le k_i\) .
